DEFINITION AND PROPERTIES OF
SUPERSOLUTIONS TO THE POROUS MEDIUM EQUATION
JUHA KINNUNEN AND PETER LINDQVIST
Abstract. We study a wide class of supersolutions of the porous medium equation. These supersolutions are defined as lower semi- continuous functions obeying the comparison principle. We show that they have a spatial Sobolev gradient and give sharp summab- ility exponents. We also study pointwise behaviour.
1. Introduction The porous medium equation
∆(um) = ∂u
∂t (1.1)
has been studied intensively during the last decades and the theory for its solutions is rather complete by now. Especially the slow diffusion casem >1 has attracted the interest of many mathematicians, because disturbances propagate with finite speed and interfaces may appear.
We refer to [15] and [14] for the theory of this fascinating equation. The objective of our work is to study a class of supersolutions, defined in an analogous way as in classical potential theory. The leading example with a singularity is the so-called Barenblatt solution, which is the fundamental solution of the porous medium equation.
The supersolutions that we have in mind are defined as lower semi- continuous functions obeying the parabolic comparison principle with respect to solutions. For lack of a better name, we have taken ourselves the liberty to call these pointwise defined functions viscosity supersolu- tions, thus distinguishing them from the ordinary supersolutions. In the stationary case the viscosity supersolutions v are exactly charac- terized by the property that the power vm is a superharmonic function, defined as in classical potential theory. In the case m= 1 the equation reduces to the heat equation and we have the supercaloric functions.
Date: February 6, 2007.
2000 Mathematics Subject Classification. 35K55.
Ordinary supersolutions are weak solutions of the inequality
∆(um)≤ ∂u
∂t
defined in the usual way with the test functions under the integral sign. Thus they are called weak supersolutions. Belonging, by defin- ition, to a parabolic Sobolev space, they are more tractable, when it comes to a priori estimates. The reader should carefully distinguish between viscosity supersolutions (Definition 3.1) and weak supersolu- tions (Definition 2.1). To this we may add that an even more restricted class of supersolutions has been treated in [12]. Among those it is the viscosity supersolutions that form a good class, closed under monotone convergence.
The most important example is a celebrated function found by Baren- blatt [3] and Zel’dovich and Kompaneets [16]. It has the formula
Bm(x, t) =
t−λ
C−λ(m−1) 2mN
|x|2 t2λ/N
1/(m−1) +
, t >0,
0, t≤0,
(1.2) where |x|2 =x21+x22 +· · ·+x2N and
λ = N
N(m−1) + 2.
The constantC >0 is at our disposal andm > 1. Heref+= max(f,0) is the positive part of f. As m → 1+ we can obtain the heat kernel.
Notice the interface (free boundary), having the equation t=c|x|N(m−1)+2.
The function Bm is, indeed, a weak solution when t > 0, but the singularity at the origin prevents Bm from being a solution in RN×R.
Strictly speaking, it is not even a weak supersolution because Z 1
−1
Z
|x|<1
|∇Bmm(x, t)|2dx dt=∞,
violating the a priori summability in the definition. However, the func- tionBmis a viscosity supersolution in the whole spaceRN×R. Needless to say, a definition that would exclude the Barenblatt solution cannot be regarded as satisfactory. As a matter of fact, the Barenblatt solution is extreme in many ways. We will utilize it to show that some results are sharp.
Our first result is that locally bounded viscosity supersolutions are weak supersolutions. Also the converse statement is true, provided the issue of semicontinuity is properly handled. We establish the existence of the spatial gradient ∇(|v|m−1v) in Sobolev’s sense. Nothing like this holds
for the time derivative, which is merely a distribution. For example, the function
v(x, t) = (
1, t >0, 0, t ≤0,
is a viscosity supersolution. Dirac’s delta function appears in the time derivative. More generally, all functions of the type v(x, t) = g(t) are viscosity supersolutions, if g(t) is a lower semicontinuous increasing function. -We have the following theorem.
Theorem 1.3. Letm≥1. Suppose thatv is a locally bounded viscosity supersolution in Ω⊂RN+1. Then the Sobolev derivatives
∂(|v|m−1v)
∂xi
, i= 1,2, . . . , N, exist and the local summability
Z t2
t1
Z
D
|∇(|v|m−1v)|2dx dt <∞ holds for each D×(t1, t2)bΩ. Moreover, we have
Z t2 t1
Z
D
∇(|v|m−1v)· ∇ϕ−v∂ϕ
∂t
dx dt≥0 whenever ϕ∈C0∞(D×(t1, t2)) and ϕ≥0.
The proof is based on a delicate approximation procedure. The approx- imants are constructed as solutions of auxiliary variational inequalities coming from a sequence of obstacle problems. The obstacles are smooth functions approximating the original function pointwise from below.
For unbounded viscosity supersolutions we can extract information by applying the theorem to the truncated functions min(v(x, t), j), j = 1,2, . . ., which are weak supersolutions. By an iterative procedure we obtain estimates which are independent of the level of truncation. The result is the theorem below, which we are content to prove for non- negative v in section 5.
Theorem 1.4. Let m ≥ 1. Suppose that v is a viscosity supersolu- tion in Ω ⊂ RN+1. Then v ∈ Lqloc(Ω), whenever 0 < q < m+ 2/N. Moreover, the Sobolev derivatives
∂(|v|m−1v)
∂xi
, i= 1,2, . . . , N, exist and the local summability
Z t2
t1
Z
D
|∇(|v|m−1v)|qdx dt <∞
holds for each D×(t1, t2)bΩ, whenever 0< q <1 + 1/(1 +mN).
The Barenblatt solution shows that the bounds for the exponents q are sharp in the theorem above. There is a reason for using themth power in the theorems. It appears even for solutions. For a nonnegative solution u it may happen that the derivative ∇(uα) does not exist in Sobolev’s sense, if 0< α < (m−1)/2. This is the case for the Baren- blatt solution. For a viscosity supersolution we have the restrictions
m−1
2 < α < m+ 1 N
on the powerα to guarantee that∇(uα) exists in Sobolev’s sense. This phenomenon is studied in the final section of the paper, where we give a Caccioppoli estimate in the above range of powers.
This phenomenon also causes a technical difficulty for the mollifications with respect to the time variable. We have to mollify |v|m−1v instead of v itself. We have found the convolution
f∗(x, t) = 1 σ
Z t 0
e(s−t)/σf(x, s)ds, σ >0,
to be very useful, in particular because its time derivative (f∗)t has a convenient form. Some technical difficulties can be concentrated in an excess term which disappears from the final estimate. The excess term is zero for smooth functions. So far, we have not been able to find any other practical way to compensate for the missing time derivative.
We have also included a section about the fine properties. While weak supersolutions are defined only almost everywhere, a distict feature of the viscosity supersolutions is that they are defined at every point in their domain. Thus the pointwise behaviour can be investigated. A central result is that, at each point in its domain, a viscosity super- solution takes the value
v(x, t) = ess lim inf
(y,τ)→(x,t),τ <tv(y, τ).
This is the content of our theorem in section 6, which is an extension of Brelot’s classical theorem for superharmonic functions. Here the notion of the essential limes inferior means that any set of (N+1)-dimensional Lebesgue measure zero can be neglected in the calculation of the lower limit.
Let us finally remark that we have deliberately decided to exclude the fast diffusion case m <1. In the linear case m = 1 our results can be read off from linear representation formulas for the heat equation. In our case the principle of superposition is not available.
Acknowledgements. Part of the research was done while the second author visited Helsinki University of Technology in September 2006.
We wish to thank the Finnish Academy of Science and Letters, the Vilho, Yrj¨o and Kalle V¨ais¨al¨a Foundation for financial support.
2. Preliminaries
This section contains some notation, definitions, and basic estimates.
Also an interesting convolution is described.
In what follows, Q will always stand for a parallelepiped Q= (a1, b1)×(a2, b2)× · · · ×(aN, bN), ai < bi, i= 1,2, . . . , N, in RN and the abbreviations
QT =Q×(0, T), Qt1,t2 =Q×(t1, t2),
where T > 0 and t1 < t2, are used for the space-time boxes in RN+1. The parabolic boundary of QT is
ΓT = (Q× {0})∪(∂Q×[0, T]).
Observe that the interior of the top Q× {T}is not included. Similarly, Γt1,t2 is the parabolic boundary of Qt1,t2. The parabolic boundary of a space-time cylinder Dt1,t2 =D×(t1, t2), whereD⊂RN is an open set, has a similar definition. In order to describe the appropriate function spaces, we recall thatH1(Q) denotes the Sobolev space of functionsu∈ L2(Q) whose first distributional partial derivatives belong to L2(Q).
The norm in H1(Q) is
kukH1(Q) =kukL2(Q)+k∇ukL2(Q).
The Sobolev space with zero boundary values, denoted byH01(Q), is the completion of C0∞(Q) with respect to the norm kukH1(Q). We denote by L2(t1, t2;H1(Q)) the space of functions such that for almost every t, t1 ≤t≤t2, the functionx7→u(x, t) belongs to H1(Q) and
Z t2
t1
Z
Q
|u(x, t)|2+|∇u(x, t)|2
dx dt < ∞.
Notice that the time derivative ut is deliberately avoided. The space L2(t1, t2;H01(Q)) is defined as the completion in the norm
Z t2
t1
Z
Q
|ϕ(x, t)|2+|∇ϕ(x, t)|2
dx dt1/2
of all functions ϕ ∈ C∞(Qt,t2) that are compactly supported in the x-variable, i.e., ϕ(·, t)∈C0∞(Q), whent1 < t < t2. In particular, ϕ= 0 on ∂Q×(t1, t2). In this space a variant of Sobolev’s inequality holds.
The parameter
κ= 1 + 1
N + 1
mM is convenient.
Lemma 2.1 (Sobolev). Suppose that u ∈ L2(t1, t2;H01(Q)). Then the inequality
Z t2 t1
Z
Q
|u|2κdx dt≤c Z t2
t1
Z
Q
|∇u|2dx dt
ess sup
t1<t<t2
Z
Q
|u(x, t)|1+1/mdx 2/N
holds with a constant cdepending only on m and N.
Proof. See Proposition 3.1 on page 7 of [5] or Chapter II §3 of [11].
To be on the safe side we give the definition of the (super)solutions of the porous medium equation, interpreted in the weak sense.
Definition 2.2. Let Ω be an open set in RN+1 and suppose that
|u|m−1u ∈ L2(t1, t2;H1(Q)) whenever Qt1,t2 b Ω. Then u is called a weak solution, if
Z t2 t1
Z
Q
∇(|u|m−1u)· ∇ϕ−u∂ϕ
∂t
dx dt= 0 (2.3) whenever Qt1,t2 b Ω and ϕ ∈ C0∞(Qt1,t2). Further, we say that u is a weak supersolution, if the integral (2.3) is non-negative for all ϕ ∈ C0∞(Qt1,t2) with ϕ≥ 0. If this integral is non-positive instead, we say that u is a weak subsolution.
Several remarks are related to the definition. The function |u|m−1u∈ L1loc(Ω) is called a distributional solution, if
Z t2
t1
Z
Q
|u|m−1u∆ϕ+u∂ϕ
∂t
dx dt= 0
whenever Qt1,t2 b Ω and ϕ ∈ C0∞(Qt1,t2). It is clear that every weak solution is a distributional solution. On the other hand, it can be shown that the converse is true as well, but the proof is more involved. The reverse heat equation is evoked in the proof. See [8] and [2].
By parabolic regularity theory the weak solutions are locally H¨older continuous, after a possible redefinition on a set of measure zero. We will need a quantitative H¨older estimate. We have locally for the oscil- lation taken over Qt1,t2 that
osc(|u|m−1u)≤c (t2−t1)α+ (diamQ)α/2
(2.4) whenever 2Q×(2t1−t2,2t2−t1)bΩ. Here α=α(N, m)>0 while c depends, in addition, on the maximum of |u| taken over the larger set 2Q×[2t1−t2,2t2−t1]. See Proposition 1.5.2 on page 65 of [14]. We also mention [4].
A continuous weak solution is called a solution. Even the spatial gradi- ent∇u of a weak solution is locally H¨older continuous. See [6] and [14]
for the regularity theory. We will not use the H¨older continuity of the gradient, but an intrinsic Harnack inequality proved by DiBenedetto is needed, see [6]. The time derivative ut has to be avoided to a cer- tain extent, because it does not necessarily exist in Sobolev’s sense. A regularization will be used to overcome this default.
If the test functionϕis required to vanish only on the lateral boundary
∂Q×[t1, t2], then the boundary terms Z
Q
u(x, t1)ϕ(x, t1)dx= lim
σ→0
1 σ
Z t1+σ t1
Z
Q
u(x, t)ϕ(x, t)dx dt
and Z
Q
u(x, t2)ϕ(x, t2)dx= lim
σ→0
1 σ
Z t2 t2−σ
Z
Q
u(x, t)ϕ(x, t)dx dt
have to be included. A direct evaluation of the integrals on the left- hand side, without the limit procedure, may occasionally yield wrong values. In the presence of discontinuities we have to pay due attention to this notation. In the case of a weak supersolution to the porous medium equation the condition becomes
Z t2 t1
Z
Q
∇(|u|m−1u)· ∇ϕ−u∂ϕ
∂t
dx dt
+ Z
Q
u(x, t2)ϕ(x, t2)dx− Z
Q
u(x, t1)ϕ(x, t1)dx≥0
(2.5)
for almost all t1 < t2 with Qt1,t2 bΩ.
The following existence result for the local Cauchy problem will be useful for us later.
Theorem 2.6. Letψ be a continuous function on the parabolic bound- ary ΓT of QT. Then there is a weak solution u ∈C(QT) of the porous medium equation in QT such that u=ψ on ΓT.
The uniqueness of the solution of the Cauchy problem follows from the comparison principle below, see [2], [8] and [14].
Theorem 2.7 (Comparison Principle). Let ψ1 and ψ2 be continuous functions on the parabolic boundary ΓT of QT such that 0≤ ψ1 ≤ ψ2. If u∈C(QT) is a weak subsolution with u=ψ1 on ΓT and v ∈C(QT) is a weak supersolution with v =ψ2 on ΓT, then u≤v in QT.
There is a principal, well-recognized difficulty with the definition. Namely, in proving estimates we usually need a test function ϕthat depends on the solution itself, for exampleϕ=uζ whereζ is a smooth cutoff func- tion. Then one cannot avoid that the “forbidden quantity” ut shows up in the calculation of ϕt. In most cases one can easily overcome
this complication by using an equivalent definition in terms of Steklov averages, as in chapter 2 of [14]. We have found the convolution
u∗(x, t) = 1 σ
Z t 0
e(s−t)/σu(x, s)ds, σ >0, (2.8) to be very useful, see page 36 in [13]. The notation hides the dependece on σ. The advantage is that no values of u(x, t) outsideQ×(0, T) are needed for the calculation of u∗(x, t) in Q×(0, T). For continuous or bounded and semicontinuous functions u the averaged function u∗ is defined at each point. We have
u∗+σ∂u∗
∂t =u. (2.9)
This implies the expedient fact that
(|u|m−1u− |u∗|m−1u∗)∂u∗
∂t ≥0 (2.10)
when m ≥1.
Some properties are listed in the following lemma.
Lemma 2.11. (i) If u∈Lp(QT), then ku∗kp,QT ≤ kukp,QT
and ∂u∗
∂t = u−u∗
σ ∈Lp(QT).
Moreover, u∗ →u in Lp(QT) as σ →0.
(ii) If, in addition,∇u ∈Lp(QT), then∇(u∗) = (∇u)∗ componentwise, k∇u∗kp,Q
T ≤ k∇ukp,Q
T, and ∇u∗ → ∇u in Lp(QT) as σ→0.
(iii) Furthermore, if uk→u in Lp(QT), then also u∗k →u∗ and ∂u∗k
∂t → ∂u∗
∂t in Lp(QT).
(iv) If ∇uk→ ∇u in Lp(QT), then ∇u∗k → ∇u∗ in Lp(QT).
(v) Analogous results hold for weak convergence in Lp(QT).
(vi) Finally, if ϕ∈C(QT), then
ϕ∗(x, t) +e−t/σϕ(x,0)→ϕ(x, t) uniformly in QT as σ→0.
Proof. The proof is rather straightforward and we leave it as an exer- cise. See [13] and [10].
The averaged equation for a weak supersolution uin Ω is the following.
If QT ⊂Ω, then Z T
0
Z
Q
∇(|u|m−1u)∗· ∇ϕ−u∗∂ϕ
∂t
dx dt
+ Z
Q
u∗(x, T)ϕ(x, T)dx
≥ Z
Q
u(x,0) 1
σ Z T
0
ϕ(x, s)e−s/σds
dx
(2.12)
for all test functionsϕ≥0 vanishing on the lateral boundary∂Q×[0, T] of QT. The reader can easily verify that (2.12) follows from (2.5). On the other hand, we obtain (2.5) from (2.12) as σ → 0. The averaged equation can also be written as
Z T 0
Z
Q
∇(|u|m−1u)∗· ∇ϕ+ϕ∂u∗
∂t
dx dt
≥ Z
Q
u(x,0) 1
σ Z T
0
ϕ(x, s)e−s/σds
dx.
(2.13)
By approximation this is valid for all non-negativeϕ∈L2(0, T;H01(Q)).
For positive weak supersolutions many a priori estimates can be derived from the simpler inequality
Z T 0
Z
Q
∇(um)∗· ∇ϕ+ϕ∂u∗
∂t
dx dt≥0 (2.14) valid for all non-negative ϕ ∈ L2(0, T;H01(Q)). We point out that (2.13) and (2.14) hold without any assumption about ϕt.
The following lemma contains a Caccioppoli type estimate. For the reader’s convenience, we give a proof of this well-known result.
Lemma 2.15 (Caccioppoli). Let |u|m−1u ∈ L2(0, T;H1(Q)) and sup- pose that |u| ≤M in QT. If u is a weak supersolution, then
Z T 0
Z
Q
ζ2|∇(|u|m−1u)|2dx dt
≤16M2mT Z
Q
|∇ζ|2dx+ 6Mm+1 Z
Q
ζ2dx
for every ζ ∈C0∞(Q) with ζ ≥0, which depends only on x.
Proof. In the averaged equation (2.13) we use the test function ϕ= (Mm− |u|m−1u)ζ2.
The crucial integral containing ∂u∗
∂t can be written as Z T
0
Z
Q
ϕ∂u∗
∂t dx dt
=Mm Z
Q
ζ2(x)u∗(x, T)dx− Z T
0
Z
Q
ζ2|u|m−1u∂u∗
∂t dx dt
=Mm Z
Q
ζ2(x)u∗(x, T)dx
− Z T
0
Z
Q
ζ2 |u|m−1u− |u∗|m−1u∗∂u∗
∂t dx dt
− Z T
0
Z
Q
ζ2|u∗|m−1u∗∂u∗
∂t dx dt.
Now we have come to a decisive point. In the second integral on the right-hand side
|u|m−1u− |u∗|m−1u∗∂u∗
∂t = |u|m−1u− |u∗|m−1u∗u−u∗
σ ≥0
since both factors have the same sign (recall that m≥1), see (2.9) and (2.10). It follows that
Z T 0
Z
Q
ϕ∂u∗
∂t dx dt≤ Z
Q
ζ2(x)
Mmu∗(x, T)− |u∗(x, T)|m+1 m+ 1
dx
and hence we obtain an estimate free of the time derivative ∂u∗
∂t . Taking this estimate into account and letting σ →0, we can write (2.13) as
Z T 0
Z
Q
∇(|u|m−1u)· ∇ϕ dx dt +
Z
Q
ζ2(x)
Mmu(x, T)− |u(x, T)|m+1 m+ 1
dx
≥ Z
Q
ζ2(x) Mmu(x,0)− |u(x,0)|m+1 dx
after some simplification. The two single integrals are not symmetric!
A deviation whent= 0 is due to the omission of the terme−t/σu(x,0) in the definition of u∗(x, t), see (vi) in Lemma 2.11. A simple estimation of the two single integrals leads to
− Z T
0
Z
Q
∇(|u|m−1u)· ∇ϕ dx dt≤3Mm+1 Z
Q
ζ2dx.
In the elliptic term we write
−∇(|u|m−1u)· ∇ϕ
=ζ2|∇(|u|m−1u)|2−2ζ∇(|u|m−1u)·(Mm− |u|m−1u)∇ζ.
The first term on the right-hand side is of the desired type. We estimate the second term on the right-hand side. The elementary inequality 2ab≤ε2a2+ε−2b2 gives
2 Z T
0
Z
Q
|ζ∇(|u|m−1u)·(Mm− |u|m−1u)∇ζ|dx dt
≤ε2 Z T
0
Z
Q
ζ2|∇(|u|m−1u)|2dx dt +ε−2
Z T 0
Z
Q
(Mm− |u|m−1u)2|∇ζ|2dx dt
≤ε2 Z T
0
Z
Q
ζ2|∇(|u|m−1u)|2dx dt+ε−2(2Mm)2T Z
Q
|∇ζ|2dx.
We choose ε= 1/
√
2 so that the first term on the right-hand side can be absorbed (the so-called Peter-Paul Principle). The result follows.
It will be crucial for us to be able to move from from one moment of time to another. The following estimate connects the future to the past. An excess term appears.
Theorem 2.16. Let v be a weak supersolution in Ω ⊂ RN+1 and Qt1,t2 bΩ. Suppose that v ≥0 and vm ∈L2(t1, t2;H01(Q)). Then
1 m+ 1
Z
Q
v(x, t2)m+1dx− 1 m+ 1
Z
Q
v(x, t1)m+1dx + lim sup
σ→0
Z t2 t1
Z
Q
(vm−(v∗)m)∂v∗
∂t dx dt +
Z t2 t1
Z
Q
|∇vm|2dx dt≥0.
Proof. Choosing ϕ=vm in (2.14) we have the basic estimate Z t2
t1
Z
Q
∇(vm)∗· ∇(vm)dx dt+ Z t2
t1
Z
Q
vm∂v∗
∂t dx dt≥0.
Since Z t2
t1
vm∂v∗
∂t dt= Z t2
t1
(v∗)m∂v∗
∂t dt+ Z t2
t1
vm−(v∗)m∂v∗
∂t dt
= 1
m+ 1 v∗(x, t2)m+1−v∗(x, t1)m+1 +
Z t2 t1
vm−(v∗)m∂v∗
∂t dt, we may safely let σ→0.
Remark 2.17. (1) Limes superior can be replaced with limes inferior, since the actual limes exists for all other terms.
(2) The excess term lim sup
σ→0
Z t2 t1
Z
Q
(vm−(v∗)m)∂v∗
∂t dx dt≥0
is not negligible. To see this, consider the essentially one-dimensional example, where v(x, t) = 0 if t ≤0 and v(x, t) = 1 if t >0. Then
v(x, t)m−v∗(x, t)m = 1− 1
σ Z t
0
e(s−t)/σds m
= 1− 1−e−t/σm
and ∂v∗
∂t (x, t) = v(x, t)−v∗(x, t)
σ = 1
σe−t/σ when t >0. Hence
Z T 0
(vm−(v∗)m)∂v∗
∂t dt= 1−e−T /σ − (1−e−T /σ)m+1 m+ 1 upon integration. Thus the excess term is
σ→0lim Z T
0
Z
Q
(vm−(v∗)m)∂v∗
∂t dx dt= m
m+ 1|Q|>0, a positive quantity.
The obstacle problem in the calculus of variations is a basic tool in our study of viscosity supersolutions. Let ψ ∈C∞(RN+1) and consider the class Fψ of all functions w∈C(QT) such that
|w|m−1w∈L2(0, T;H1(Q)), w=ψ on ΓT, and w≥ψ inQT. The function ψ acts as an obstacle and also prescribes the boundary values.
The following existence theorem will be useful for us later.
Lemma 2.18. There is a unique w∈ Fψ such that Z T
0
Z
Q
∇(|w|m−1w)· ∇(φ−w) + (φ−w)∂φ
∂t
dx dt
≥ 1 2
Z
Q
|φ(x, T)−w(x, T)|2dx
(2.19)
for all smooth functions φ in the class Fψ. In particular, w is a con- tinuous weak supersolution. Moreover, in the open set {w > ψ} the function w is a solution.
Proof. The existence can be shown as in the proof of Theorem 3.2 in [1]. Continuity follows from standard regularity theory, but it seems to be difficult to find a convenient reference.
3. Viscosity supersolutions
In this section we define the class of viscosity supersolutions and prove Theorem 1.3. We need approximating weak supersolutions. They are constructed via an obstacle problem in the calculus of variations. The procedure will be discussed below.
Let us begin with the definition, which is similar to the classical defin- ition of superharmonic functions, due to F. Riesz. We remark once more that the word “viscosity” is only used as a label by us.
Definition 3.1. A function v : Ω → (−∞,∞] is called a viscosity supersolution if
(1) v is lower semicontinuous in Ω, (2) v is finite in a dense subset of Ω,
(3) v satisfies the following comparison principle in each subdomain Dt1,t2 b Ω: if h ∈ C(Dt1,t2) is a solution in Dt1,t2 and if h ≤v on the parabolic boundary of Dt1,t2, thenh ≤v in Dt1,t2. It follows immediately that the pointwise minimum
v(x, t) = min(v1(x, t), v2(x, t), . . . , vj(x, t))
of finitely many viscosity supersolutions is a viscosity supersolution.
Another useful construction for a non-negative viscosity supersolution v is to redefine it as 0 till a given instant t0. In other words
v0(x, t) =
(0, t ≤t0, v(x, t), t > t0,
is a viscosity supersolution. In section 5 a further construction, the so-called Poisson modification, is discussed.
Notice that a viscosity supersolution is defined at every point in its domain. No differentiability is presupposed in the definition. The only tie to the differential equation is through the comparison principle.
It turns out that a viscosity supersolution satisfies the comparison prin- ciple in more general domains than the cylinders Dt1,t2. For our pur- poses it is sufficient that the comparison principle holds for a finite union of boxes. The proof is a matter of successive comparisons, start- ing with the earliest boxes.
There is a relation between weak supersolutions and viscosity super- solutions. Roughly speaking, the weak supersolutions are viscosity su- persolutions, provided the issue about lower semicontinuity is properly
handled. In particular, a continuous supersolution is a viscosity super- solution. On the other hand, a bounded viscosity supersolution is a weak supersolution.
The Barenblatt solution clearly shows that the class of viscosity su- persolutions contains more than weak supersolutions. Nevertheless, it turns out that a viscosity supersolution can be approximated point- wise with an increasing sequence of weak supersolutions, constructed through successive obstacle problems. Let us describe this procedure.
Theorem 3.2. Suppose that v is a viscosity supersolution in Ω and let Qt1,t2 b Ω. Then there is a sequence of weak supersolutions vk ∈ C(Qt1,t2),
|vk|m−1vk∈L2(t1, t2;H1(Q)), k = 1,2, . . . , such that
v1 ≤v2 ≤ · · · ≤v
and vk →v pointwise in Qt1,t2 as k → ∞. If, in addition, v is locally bounded in Ω, then |v|m−1v ∈ L2(t1, t2;H1(Q)) and v itself is a weak supersolution.
Proof. The lower semicontinuity implies that there is a sequence of functions ψk ∈C∞(Ω), k= 1,2, . . ., such that
ψ1 < ψ2 < . . . and lim
k→∞ψk =v
at every point of Ω. It is decisive here that the inequality ψk < ψk+1
is strict. Next, using the functions ψk as obstacles, we construct su- persolutions of (1.1) that approximate v from below. This has to be done locally, say in a given box Qt1,t2 with Qt1,t2 bΩ. To simplify the notation we consider QT, assuming that QT bΩ. Letvk ∈C(QT),
|vk|m−1vk∈L2(0, T;H1(Q)), k = 1,2, . . . ,
denote the solution of the obstacle problem in QT with the obstacle ψk, see Lemma 2.18. Thus vk ∈ Fψ
k. We claim that
v1 ≤v2 ≤. . . and vk≤v, k= 1,2, . . . ,
in QT. In particular, ψk ≤ vk ≤ v then gives the desired convergence.
Due to a technical difficulty, we choose an arbitrarily small ε >0 and prove that vk(x, t)≤v(x, t) when 0< t < T −ε and x∈Q. The set
Kk ={(x, t) :x∈Q, 0≤t≤T −ε, vk(x, t)≥ψk+1(x, t)}, k = 1,2, . . ., is compact. The distance ofKkto the set wherevk(x, t) = ψk(x, t) is positive, sayδ =δ(k, ε). (We assume that the set Kk is not empty.) This is due to the continuity of the functions and the strict inequality ψk+1 > ψk. Recall that in the set where vk(x, t) > ψk(x, t)
the function vk is a solution of the porous medium equation. Now we want to use the comparison principle.
Suppose that RN+1 has been divided into dyadic cubes in the standard way. The set Kk can be covered with a finite number of (closed) dyadic cubes, all of the same size and with a sufficiently small diameter, say that the diameter of each cube is smaller than δ/2. The interior of the union of the cubes has its parabolic boundary in the set where ψk < vk ≤ ψk+1. Thus vk ≤ ψk+1 < v on this parabolic boundary.
By the comparison principle we conclude that vk ≤ v at least in Kk. Outside Kk the inequality vk < ψk+1 < v holds trivially. This shows that vk ≤v inQT since ε >0 was arbitrary. The inequality vk ≤vk+1
can be shown in the same way.
If v is locally bounded, a compactness argument is available. The Caccioppoli estimate (Lemma 2.15) gives
Z T 0
Z
Q
ζ2|∇(|vk|m−1vk)|2dx dt
≤16M2mT Z
Q
|∇ζ|2dx+ 6Mm+1 Z
Q
ζ2dx,
whereζ depends only on x,ζ ∈C0∞(Q),ζ ≥0 andM is the supremum of v in the support of ζ. By weak compactness, ∇(|v|m−1v) exists in Sobolev’s sense and
∇(|vk|m−1vk)→ ∇(|v|m−1v)
weakly in L2 ask → ∞. Hence we may proceed to the limit under the integral sign in
Z T 0
Z
Q
∇(|vk|m−1vk)· ∇ϕ−vk
∂ϕ
∂t
dx dt≥0, where ϕ∈C0∞(QT). This shows that v is a weak supersolution.
Theorem 1.3 follows immediately from the previous theorem.
We seize the opportunity to mention that the following result can be extracted from the end of the previous proof.
Proposition 3.3. Consider a sequence v1 ≤v2 ≤. . .
of weak supersolutions in QT such that |vk| ≤ M and |vk|m−1vk ∈ L2(0, T;H1(Q)) when k = 1,2, . . . Then also the limit function
v = lim
k→∞vk
is a weak supersolution. In particular, |v|m−1v ∈ L2(0, T;H1(Q0)) whenever Q0 ⊂⊂Q.
The following Harnack type convergence theorem holds for solutions of the porous medium equation. We need a continuous limit function directly, without having to correct the function in a set of measure zero.
Lemma 3.4 (Harnack). Suppose that hk, k = 1,2, . . ., is a sequence of weak solutions in Ω and that 0 ≤h1 ≤ h2 ≤ · · · pointwise in Ω. If the limit function
h(x, t) = lim
k→∞hk(x, t)
is finite in a dense subset of Ω, then h is a weak solution inΩ. If each hk is continuous, so is h.
Proof. The intrinsic Harnack inequalities proved by DiBenedetto in [6]
can be passed over from the sequence hk, k = 1,2, . . ., to the limit function h. It follows that h is locally bounded. Then we may use the compactness argument at the end of the proof of Theorem 3.2 to conclude that the limit of the equations
Z Z
Ω
∇hmk · ∇ϕ−hk
∂ϕ
∂t
dx dt= 0
as k→0 is the required equation for h. Here ϕ∈C0∞(Ω). This proves that h is a weak solution.
Suppose that each hk is continuous. Since 0 ≤ hk ≤ h and h is loc- ally bounded, the local H¨older estimate (2.4) for osc(hmk) holds with a constant cwhich is independent of the index k. Thus the family{hmk } is locally equicontinuous. The continuity of hm follows from Ascoli’s theorem. This implies that h is continuous.
4. Preliminary summability estimates
This section is devoted to some technical estimates, where the functions vm have to be truncated at the level k (the original functions v at the level k1/m). According to Theorem 3.2 the truncated functions are supersolutions. The notation
wm = (vm)j = min(v(x, t)m, j) will be used for a large index j. We also write
(wm)k = min(vm, k)
for k = 0,1, . . . , j. A test function used by Kilpel¨ainen and Mal´y in [9]
will play an essential role.
Lemma 4.1. Let m >1 and let Ω⊂RN+1 be a domain with QT bΩ.
Suppose that v ≥ 0 is a viscosity supersolution in Ω, min(vm, j) ∈ L2(0, T;H01(Q)) and v(x,0) = 0 when x ∈ Q. Let j be a positive integer and denote wm = (vm)j = min(vm, j). Then
Z T 0
Z
Q
|∇(wm)|2dx dt+ 1 m+ 1
Z
Q
wm+1dx
+ lim sup
σ→0
Z T 0
Z
Q
wm−(w∗)m∂w∗
∂t dx dt
≤j Z T
0
Z
Q
|∇(wm)1|2dx dt+ lim sup
σ→0
Z T 0
Z
Q
(wm)1
∂w∗
∂t dx dt
.
Remark 4.2. Observe that the familiar excess term appears on the left- hand side of the estimate. It is needed to counterbalance the excess term in Theorem 2.16.
Proof. Choose the test function ϕ= (wm)k−(wm)k−1
− (wm)k+1−(wm)k
,
where k= 1,2, . . . , j−1. Now ϕ≥0. Since wis a weak supersolution, we have
Z T 0
Z
Q
∇(wm)∗· ∇ϕ dx dt+ Z T
0
Z
Q
ϕ∂w∗
∂t dx dt≥0 according to (2.14). This implies that
Z T 0
Z
Q
∇(wm)∗· ∇ (wm)k+1−(wm)k
dx dt
+ Z T
0
Z
Q
(wm)k+1−(wm)k
∂w∗
∂t dx dt
≤ Z T
0
Z
Q
∇(wm)∗ · ∇ (wm)k−(wm)k−1
dx dt
+ Z T
0
Z
Q
(wm)k−(wm)k−1
∂w∗
∂t dx dt.
for every k = 1,2, . . . , j−1. We abbreviate the previous expression as ak+1 ≤ak
from which it follows, by recursion, that Xj
k=1
ak ≤ja1. (4.3)
The notation hides the dangerous fact that ak depends onj. Because of cancellation the sum on the left-hand side of (4.3) can be computed
as Xj
k=1
ak= Z T
0
Z
Q
∇(wm)∗· ∇(wm)jdx dt+ Z T
0
Z
Q
(wm)j
∂w∗
∂t dx dt, where (wm)j =wm and consequently
(wm)j
∂w∗
∂t =wm∂w∗
∂t = wm−(w∗)m∂w∗
∂t + (w∗)m∂w∗
∂t . A simple integration gives
Z T 0
Z
Q
(w∗)m∂w∗
∂t dx dt= 1 m+ 1
Z
Q
(w∗)m+1dx since w∗(x,0) = 0. This implies that
Xj k=1
ak = Z T
0
Z
Q
∇(wm)∗· ∇wmdx dt +
Z T 0
Z
Q
wm−(w∗)m∂w∗
∂t dx dt+ 1 m+ 1
Z
Q
(w∗)m+1dx.
The right-hand side of (4.3) is ja1 =j Z T
0
Z
Q
∇(wm)∗ · ∇(wm)1dx dt+ Z T
0
Z
Q
(wm)1
∂w∗
∂t dx dt .
Thus we have Z T
0
Z
Q
∇(wm)∗ · ∇wmdx dt+ Z T
0
Z
Q
wm−(w∗)m∂w∗
∂t dx dt
+ 1
m+ 1 Z
Q
(w∗)m+1dx
≤j Z T
0
Z
Q
∇(wm)∗·∇(wm)1dx dt+ Z T
0
Z
Q
(wm)1
∂w∗
∂t dx dt
.
The claim follows from this letting the smoothing parameter σ→0.
Let us proceed a little further under the same assumptions. Let 0 <
t1 < T and t1 ≤τ ≤T. By Theorem 2.16 we have 1
m+ 1 Z
Q
w(x, t)m+1dx
≤ Z τ
0
Z
Q
|∇(wm)|2dx dt+ 1 m+ 1
Z
Q
w(x, τ)m+1dx + lim sup
σ→0
Z τ 0
Z
Q
(wm−(w∗)m)∂w∗
∂t dx dt
for every 0< t < t1. Together with Lemma 4.1 this implies that
ess sup
0<t<t1
1 m+ 1
Z
Q
w(x, t)m+1dx
≤ Z τ
0
Z
Q
|∇(wm)|2dx dt+ 1 m+ 1
Z
Q
w(x, τ)m+1dx + lim sup
σ→0
Z τ 0
Z
Q
(wm−(w∗)m)∂w∗
∂t dx dt
≤j Z τ
0
Z
Q
|∇(wm)1|2dx dt+ lim sup
σ→0
Z τ 0
Z
Q
(wm)1
∂w∗
∂t dx dt
.
Using Lemma 4.1 again we conclude that
ess sup
0<t<t1
1 m+ 1
Z
Q
w(x, t)m+1dx+ Z t1
0
Z
Q
|∇(wm)|2dx dt
≤2j Z τ
0
Z
Q
|∇(wm)1|2dx dt + lim sup
σ→0
Z τ 0
Z
Q
(wm)1
∂w∗
∂t dx dt .
(4.4)
Observe that the excess term disappeared.
The estimation of the right-hand side of (4.4) is postponed till section 5. We proceed by assuming, for the moment, that we already have archieved the bound
ess sup
0<t<t1
1 m+ 1
Z
Q
w(x, t)m+1dx+ Z t1
0
Z
Q
|∇(wm)|2dx dt
≤cj Z
Q
w(x, τ)dx+T|Q|
.
(4.5)
We integrate both sides of this inequality with respect to τ over the interval [t1, T] and divide by T −t1. This implies
ess sup
0<t<t1
1 m+ 1
Z
Q
w(x, t)m+1dx+ Z t1
0
Z
Q
|∇(wm)|2dx dt
≤cj 1 T −t1
Z T 0
Z
Q
w(x, t)dx dt+ T T −t1
|Q| .
Recall thatwm = min(vm, j). Hence the previous estimate implies that ess sup
0<t<t1
1 m+ 1
Z
Q
min(vm, j)1+1/mdx +
Z t1
0
Z
Q
|∇min(vm, j)|2dx dt
≤cj 1 T −t1
Z T 0
Z
Q
min(vm, j)1/mdx dt+ T|Q| T −t1
≤cj T|Q| T −t1
j1/m+ T|Q| T −t1
≤cj1+1/m T|Q| T −t1
.
(4.6)
The constant c may change from one line to the next.
So far the index j has been kept fixed. Now we assume that min(vm, j) belongs to the space L2(0, T;H01(Q)) for each j = 1,2, . . . We aim at estimating the measure of the sets
Ej =
(x, t)∈Qt1 :j ≤vm(x, t)<2j
for j = 1,2, . . . Let κ = 1 + 1/N + 1/(mN). Sobolev’s inequality (Lemma 2.1) and (4.6) imply that
j2κ|Ej| ≤ ZZ
Ej
(min(vm,2j))2κdx dt
≤ Z t1
0
Z
Q
(min(vm,2j))2κdx dt
≤c Z t1
0
Z
Q
|∇min(vm,2j)|2dx dt
·
ess sup
0<t<t1
Z
Q
min(vm,2j)1+1/mdx 2/N
≤cj(1+1/m)(1+2/N) T|Q| T −t1
1+2/N
,
for j = 1,2, . . . It follows that
|Ej| ≤cj−1+1/m
T|Q| T −t1
1+2/N
for j = 1,2, . . . Letα >0. From this we conclude that the sum in Z t1
0
Z
Q
vmαdx dt≤T|Q|+ X∞
j=1
ZZ
E2j−1
vmαdx dt
can be majorized by X∞
j=1
ZZ
E2j−1
vmαdx dt≤ X∞
j=1
2αj|E2j−1|
≤c
T|Q| T −t1
1+2/nX∞ j=1
2−j(−α+1−1/m)
.
The series converges if α < 1−1/m. Thus we have a finite majorant.
Indeed, since m >1 there is a small α1 >0 such that Z t1
0
Z
Q
vmα1dx dt <∞.
We may assume that 0< α1 <1/m.
This was the first step. In order to improve the exponent we iterate this procedure. At the next step we split 1/m as
1
m =α1+ 1
m −α1
. Let 0 < t2 < t1. As in (4.6) we have
ess sup
0<t<t2
1 m+ 1
Z
Q
min(vm, j)1+1/mdx +
Z t2
0
Z
Q
|∇min(vm, j)|2dx dt
≤cj 1 t1−t2
Z t1
0
Z
Q
min(vm, j)1/mdx dt+ t1|Q| t1−t2
≤cj 1
t1−t2
Z t1 0
Z
Q
min(vm, j)α1min(vm, j)1/m−α1dx dt + t1|Q|
t1−t2
≤cj1+1/m−α1 1 t1−t2
Z t1
0
Z
Q
vmα1dx dt+ t1|Q| t1−t2
.
The right-hand side of this estimate is a finite number by the first step of the iteration. Hence we may apply Sobolev’s inequality and estimate the size of the distribution set Ej as before. It follows that
|Ej∩Qt2| ≤cj−1+1/m−α1(1+2/N)
for j = 1,2, . . . where the constant c depends on various parameters.
From this we conclude that Z t2
0
Z
Q
vmα2dx dt <∞
whenever
α2 <1− 1 m +α1
1 + 2
N
.
Certainly α2 > α1. At each step of the iteration we obtain αk+1 > ak. After a finite number of steps we reach 1/m and here we stop. We choose k such that αk < 1/m ≤ ak+1. We have now reached at least all exponents α < 1/m. At the final step of the iteration we use that ak< α <1/m from which we conclude that
Z tk+1 0
Z
Q
vmαdx dt <∞
whenever
α <1 + 2 mN. Indeed, the exponent is the supremum of all
1− 1 m +α
1 + 2 N
,
where α < 1/m. To reach a given exponent α only a finite number of steps is needed and hence the influence of the tk’s is under control.
We have proved the following result, in which the correct exponent is present.
Theorem 4.7. Letm >1and letΩ⊂RN+1 be a domain withQT bΩ.
Suppose that v ≥ 0 is a viscosity supersolution in Ω, min(vm, j) ∈ L2(0, T;H01(Q)) and v(x,0) = 0 when x ∈ Q. Let j be a positive integer and denote wjm = (vm)j = min(vm, j). Fix t1 < T. If there is a constant c such that
ess sup
0<t<t1
1 m+ 1
Z
Q
wj(x, t)m+1dx+ Z t1
0
Z
Q
|∇(wjm)|2dx dt
≤cj Z
Q
wj(x, τ)dx+T|Q|
,
when t1 < τ < T, then vm ∈ Lq(Qt∗) when t∗ < t1 for every 0 < q <
1 + 2/(mN).
Let us continue the reasoning turning our attention to the summability of the gradient. Let ε >0 be small and fixt∗ < t1. The two estimates
Z t∗ 0
Z
Q
|∇wjm|dx dt≤cεj1+ε
and
|Ej∗| ≤cεj−(1+2/(mN))+ε,
where Ej∗ = Ej ∩Qt∗ (thus the instances t > t∗ are excluded), can be read off from the previous considerations. Here cε depends also
